Coalgebras occur naturally in a number of contexts (for example, group schemes).
(Here and refer to the tensor product over K.)
Equivalently, the following two diagrams commute:
In the first diagram we silently identify with ; the two are naturally isomorphic. Similarly, in the second diagram the naturally isomorphic spaces , and are identified.
The first diagram is the dual of the one expressing associativity of algebra multiplication (called the coassociativity of the comultiplication); the second diagram is the dual of the one expressing the existence of a multiplicative identity. Accordingly, the map Δ is called the comultiplication (or coproduct) of C and ε is the counit of C.
Take an arbitrary set S and form the K-vector space with basis S. The elements of this vector space are those functions from S to K that map all but finitely many elements of S to zero; we identify the element s of S with the function that maps s to 1 and all other elements of S to 0. We will denote this space by C. We define
for all Again, because of linearity, this suffices to define Δ and ε uniquely on all of K[X]. Now K[X] is both a unital associative algebra and a coalgebra, and the two structures are compatible. Objects like this are called bialgebras, and in fact most of the important coalgebras considered in practice are bialgebras. Examples include Hopf algebras and Lie bialgebras.
In finite dimensions, the duality between algebras and coalgebras is closer: the dual of a finite-dimensional (unital associative) algebra is a coalgebra, while the dual of a finite-dimensional coalgebra is a (unital associative) algebra. Indeed in general, the dual of a coalgebra is an algebra, but the dual of an algebra is not always a coalgebra.
The key point is that in finite dimensions, .
To distinguish these: in general, algebra and coalgebra are dual notions (meaning that their axioms are dual: reverse the arrows), while for finite dimensions, they are dual objects (meaning that a coalgebra is the dual object of an algebra and conversely).
If A is a finite-dimensional unital associative K-algebra, then its K-dual A* consisting of all K-linear maps from A to K is a coalgebra. The multiplication of A can be viewed as a linear map , which when dualized yields a linear map . In the finite-dimensional case, is naturally isomorphic to , so we have defined a comultiplication on A*. The counit of A* is given by evaluating linear functionals at 1.
When working with coalgebras, a certain notation for the comultiplication simplifies the formulas considerably and has become quite popular. Given an element c of the coalgebra (C,Δ,ε), we know that there exist elements c(1)(i) and c(2)(i) in C such that
The fact that ε is a counit can then be expressed with the following formula
The coassociativity of Δ can be expressed as
Some authors omit the summation symbols as well; in this sumless Sweedler notation, we may write
A coalgebra is called co-commutative if ;, where is the K-linear map defined by for all c,d in C. In Sweedler's sumless notation, C is co-commutative if and only if
If and are two coalgebras over the same field K, then a coalgebra morphism from to is a K-linear map such that and . In Sweedler's sumless notation, the first of these properties may be written as:
A subspace D of C is called a subcoalgebra if Δ(D)⊆D⊗D; in that case, D is itself a coalgebra, with the restriction of ε to D as counit.
The kernel of every coalgebra morphism f : C1 → C2 is a coideal in C1, and the image is a subcoalgebra of C2. The common isomorphism theorems are valid for coalgebras, so for instance C1/ker(f) is isomorphic to im(f).
As we have seen above, if A is a finite-dimensional unital associative K-algebra, then A* is a finite-dimensional coalgebra, and indeed every finite-dimensional coalgebra arises in this fashion from some finite-dimensional algebra (namely from the coalgebra's K-dual). Under this correspondence, the commutative finite-dimensional algebras correspond to the cocommutative finite-dimensional coalgebras. So in the finite-dimensional case, the theories of algebras and of coalgebras are dual; studying one is equivalent to studying the other. However, things diverge in the infinite-dimensional case: while the K-dual of every coalgebra is an algebra, the K-dual of an infinite-dimensional algebra need not be a coalgebra.
Every coalgebra is the sum of its finite-dimensional coalgebras, something that's not true for algebras. In a certain sense then, coalgebras are generalizations of (duals of) finite-dimensional unital associative algebras.
Corresponding to the concept of representation for algebras is a corepresentation of a coalgebra. The notion is dual to the usual algebra representation as follow: An n-dimensional (left) corepresentation of C is given by a map where V is an n-dimensional vector space, such that it satisfies:
It can be represented as where is a basis of V, and is a matrix with elements in C (i.e. ). The conditions above implies that